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Mathematics Department, UCLA
T. Richthammer
spring 09, midterm 1
Apr 27, 2009
Midterm 1: Math 170B Probability, Sec. 1
1. Let
Z
=
X
+
Y
, where
X, Y
are independent RVs with uniform distribution on [0
,
1].
(a) Calculate
E
(
Z
),
V
(
Z
) and the PDF of
Z
.
(b) State for which of the results in (a) you really need the independence of
X, Y
.
Answer:
(a)
E
(
X
) =
R
1
0
xdx
=
1
2
and
E
(
X
2
) =
R
1
0
x
2
dx
=
1
3
, i.e.
V
(
X
) =
1
3

1
4
=
1
12
. So
E
(
Z
) =
E
(
X
) +
E
(
Y
) = 1 and
V
(
Z
) =
V
(
X
) +
V
(
Y
) =
1
6
.
Using the formula from the lecture the PDF
f
of
Z
is
f
(
z
) =
R
f
X
(
x
)
f
Y
(
z

x
)
dx
=
R
1
{
0
≤
x
≤
1
,
0
≤
z

x
≤
1
}
dx
=
R
1
{
max
{
0
,z

1
}≤
x
≤
min
{
1
,z
}}
dx
, which is =
R
z
0
dx
=
z
for 0
≤
z
≤
1
and =
R
1
z

1
dx
= 2

z
for 1
≤
z
≤
2, so
f
Z
(
z
) =
z
1
{
0
≤
z
≤
1
}
+ (2

z
)1
{
1
<z
≤
2
}
.
Without using the formula from the lecture: For 0
≤
c
≤
2 we have
F
Z
(
c
) =
P
(
Z
≤
c
) =
λ
2
(
A
c
), where
A
c
is the subset of a square below the line
y
=
c

x
. For
c
≤
1 we have
λ
2
(
A
c
) =
1
2
c
2
, and for
c
≥
1 we have
λ
2
(
A
c
) = 1

1
2
(2

c
)
2
. So
f
Z
(
c
) =
F
±
Z
(
c
) =
c
for
c
≥
1 and = 2

c
for
c >
1, and we get the same answer as above.
(b) Independence is needed for the variance and the PDF, but not for the expectation.
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 Winter '10
 RICHTHAMMER

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